Mathematics Project(2)<The Nine-point Circle>4B 12, 23, 37, 38, 39, 41

1. Basic Concepts

1a. Related knowledge: Perpendicular feet

The point on the leg

opposite a given vertex

of a triangle at which

the perpendicular

passing through that

vertex intersects the

side.

1a. Related knowledge: Orthocenter

The interjection of the

three perpendicular

feet(or altitudes) of a

triangle.

1a. Related knowledge Centroid (of a triangle) The intersecting point

of three medians of a

triangle.

Median: A line

segment drawn from

one vertex to the

midpoint of the

opposite side.

1b. Definition

The circle has 9 points on the circumference, which are: Three mid-points of the

sides of a triangle(M); Three perpendicular feet

drawn from the vertices of the triangle to the nearest base(H); and

The mid-points of the segments that join the vertices and the orthocenter(E).

2. History of Nine-Point Circle

2a. First raise of the Nine-Point Circle

Benjamin Beven, a English mathematician ,

first raised the problem of nine-point circle in

an English magazine in 1804.

2b. First being proved

The first person who proved the problem

raised by Benjamin Beven is reasonable was

Charles Brainchon, a French mathematician

There is another claim that the prove was

done by Joseph Gergonne and Brainchon.

2c. Raised of Properties of Nine-Point Circle A German geometer,

Karl Wilhelm Feuerbach has investigated about the nine-point circle too.

《直邊三角形的一些特殊點的性質》 Including some

important properties of nine-point circle

3. Properties of the Nine-Point Circle

3a. The Radius

The radius of the nine-point circle is half of the radius of its circumradius.

Circumcircle: For any triangle there is always a circle passing through its three vertices. Circumradius: Radius

of circumcircle

3b. The Center

The center of the nine-point circle is the midpoint of the Euler line.

Euler line: A line which is produced by joining the orthocenter and the centroid of a triangle.

3c. The Feuerbach’s Theorem

Incircle: The unique circle

that is tangent to each of

the triangle's three sides.

Excircle: The circle

tangent to the extended

two nonadjacent sides of a

triangles and to the other

side of the triangle.

3c. The Feuerbach’s Theorem(Continue)

The nine-point circle of

a triangle “touches” the

incircle and the three

excircles.

3d. Identical nine-point circle

All triangles inscribed in a

given circle (to

circumscribe) and having

the same orthocenter, have

the same nine-point circle.

Circumscribe: The

vertices of the polygon are

on the circumference of the

circle.

The End

Thank you!